Problems Optimization

1. Minimum of the Rosenbrock function

   1. Find, using one of the minimization routines from GSL, the minimum of the Rosenbrock function,

      f(x,y) = (1-x)² + 100(y-x²)²
      

   2. Print out the number of iterations.
   3. Print out the points where the algorithm estimated the function.
   4. (Optional) Illustrate the path the algorithm took toward the minimum.

2. Least-squares fit

   Suppose an experiment measuring the activity of a radioactive substance as function of time gives the following result,

   # t_i   y_i   e_i
    0.47  5.49  0.26
    1.41  4.08  0.12
    2.36  3.54  0.27
    3.30  2.61  0.10
    4.24  2.09  0.15
    5.18  1.91  0.11
    6.13  1.55  0.13
    7.07  1.47  0.07
    8.01  1.45  0.15
    8.95  1.25  0.09
   

   or

   double t[]= {0.47,1.41,2.36,3.30,4.24,5.18,6.13,7.07,8.01,8.95};
   double y[]= {5.49,4.08,3.54,2.61,2.09,1.91,1.55,1.47,1.45,1.25};
   double e[]= {0.26,0.12,0.27,0.10,0.15,0.11,0.13,0.07,0.15,0.09};
   int n = sizeof(t)/sizeof(t[0]);
   

   where the time t_i, activity y_i, and the experimental error σ_i are given in some rescaled units.

   Now,

   1. Do the least-squares fit of the function

      f(t)=A*exp(-t/T)+B
      

      (where the initial amount A, the lifetime T, and the background noise B are the fitting parameters) to the data and estimate the lifetime, T, of the substance.
   2. Plot the experimental data and the fitting function.
   3. (Optional) Do the same least-squares fit with another algorithm from gsl_multimin.h and find out which make fewer evaluations of the master function.
   4. (Optional) Do the fit using the Gnuplot's built-in fitting command fit and compare with your fit.

   Hints:

   1. It is probably easier to use the algorithm without derivatives gsl_multimin_fminimizer_nmsimplex2.

   2. The (master) function to minimize is F(A,T,B)=∑i(f(ti)-yi)²/σi². That is, you have a three-dimensional minimization problem in the space of the parameters A, T, and B.

      It can be programmed like this,

      struct experimental_data {int n; double *t,*y,*e;};
      
      double function_to_minimize (const gsl_vector *x, void *params) {
      	double  A = gsl_vector_get(x,0);
      	double  T = gsl_vector_get(x,1);
      	double  B = gsl_vector_get(x,2);
      	struct experimental_data *p = (struct experimental_data*) params;
      	int     n = p->n;
      	double *t = p->t;
      	double *y = p->y;
      	double *e = p->e;
      	double sum=0;
      	#define f(t) A*exp(-(t)/T) + B
      	for(int i=0;i<n;i++) sum+ = pow( (f(t[i]) - y[i])/e[i] ,2);
      	return sum;
      }